Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves x=4y2−y3 and x=0 about the x-axis.

Answer: Satisfied for n=1, n=k and n=k+1

Step-by-step explanation:

The induction procedure involves two steps

First is

Basic Step

Here we consider that for the value n=1, there is one car and it will always make the full circle.

Induction Step

Since basic step is satisfied for n=1

Now we do it for n=k+1

Now according to the statement a car makes full circle by taking gas from other cars as it passes them. This means there are cars that are there to provide fuel to the car. So we have a car that can be eliminated i.e. it gives it fuels to other car to make full circle so it is always there.

Now ,go through the statement again that the original car gets past the other car and take the gas from it to eliminate it. So now cars remain k instead of k+1 as it's fuel has been taken. Now the car that has taken the fuel can make the full circle. The gas is enough to make a circle now.

So by induction we can find a car that satisfies k+1 induction so for k number of cars, we can also find a car that makes a full circle.

Answer:

Number of student = 25

Step-by-step explanation:

Let x be the number of student and y be the others

A new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else

[tex]x+y=55\\y=55-x[/tex]

[tex]5x+10y=425[/tex]

replace y with 55-x

[tex]5x+10y=425\\5x+10(55-x)=425\\5x+550-10x=425\\-5x+550= 425[/tex]

Subtract 550 from both sides

[tex]-5x+550= 425\\-5x= -125\\x=25[/tex]

[tex]y=55-x\\y=55-25\\y=30\\[/tex]

Number of student = 25

Answer: 25 students used the service this month.

Step-by-step explanation:

Let x represent the number of students that used the streaming service this month.

Let y represent the number of people apart from students that used the streaming service this month.

This month, the service had 55 users. It means that

x + y = 55

The new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else. They collected a total of 425 dollars. It means that

5x + 10y = 425 - - - - - - -1

Substituting x = 55 - y into equation 1, it becomes

5(55 - y) + 10y = 425

275 - 5y + 10y = 425

- 5y + 10y = 425 - 275

5y = 150

y = 150/5 = 30

x = 55 - y = 55 - 30

x = 25

Answer:

116,280 ways

Step-by-step explanation:

The number of ways of assigning the accounts to each of the salesperson is computed by combination

Number of ways = n combination r = n!/(n-r)!r!

n = 21, r = 7

Number of ways = 21 combination 7 = 21!/(21-7)!7! = 21!/14!7! = 116,280 ways

Answer:

N = 6300 ways

She can choose the materials 6300 ways

Step-by-step explanation:

In this case order of selection is not important, so we use combination.

For solids,

She needs 2 out of 9 available solids = 9C2

For striped prints

She needs 4 out of 7 available = 7C4

For floral prints

She needs 4 out of 5 available = 5C4

The total number of ways she can choose the materials is;

N = 9C2 × 7C4 × 5C4

N = 9/(7!2!) × 7/(4!3!) × 5/(4!1!)

N = 6300 ways

To find the number of different ways Courtney can choose the materials for her quilt, we can use the concept of combinations. The total number of ways she can choose the materials is the product of the number of choices for each type of fabric.

To find the number of different ways Courtney can choose the materials for her quilt, we can use the concept of combinations. The total number of ways she can choose the materials is given by the product of the number of choices for each type of fabric. So, the answer is:

Total number of ways = number of ways to choose solids * number of ways to choose floral prints * number of ways to choose striped fabrics

Given that she needs 2 solids, 4 floral prints, and 4 striped fabrics, we can calculate:

Substituting these values into the formula:

Total number of ways = 36 * 5 * 35 = 6300

So, there are 6300 different ways Courtney can choose the materials for her quilt.

Answer:

No, the standard deviation for number of pages in a romance novel is six only.

Step-by-step explanation:

First we state our Null Hypothesis, [tex]H_o[/tex] : [tex]\sigma[/tex] = 6

             and Alternate Hypothesis, [tex]H_1[/tex] : [tex]\sigma[/tex] > 6

We have taken these hypothesis because we have to check whether our population standard deviation is higher than what editor hypothesized of 6 pages in a romance novel.

Now given sample standard deviation, s = 9 and sample size, n = 25

To test this we use Test Statistics = [tex]\frac{(n-1)s^{2} }{\sigma^{2} }[/tex] follows chi-square with (n-1) degree of freedom [[tex]\chi ^{2}_n__-1[/tex]]

       Test Statistics = [tex]\frac{(25-1)9^{2} }{6^{2} }[/tex] follows [tex]\chi ^{2}_2_4[/tex]  = 54

and since the level of significance is not stated in question so we assume it to be 5%.

Now Using chi-square table we observe at 5% level of significance the [tex]\chi ^{2}_2_4[/tex] will give value of 36.42 which means if our test statistics will fall below 36.42 we will reject null hypothesis.

Since our Test statistics is more than the critical value i.e.(54>36.42) so we have sufficient evidence to accept null hypothesis and conclude that our population standard deviation is not more than 6 pages which the editor hypothesized.

Approximately 68.26% of the measurements fall between 46 and 74 in this distribution.

To find the proportion of measurements between 46 and 74 in a normal distribution with a mean (μ) of 60 and a standard deviation (σ) of 14, we can use the standard normal distribution (z-score) and the cumulative distribution function (CDF).

First, we need to convert the interval endpoints to z-scores using the formula:

z = (x - μ) / σ

Where x is the value in the interval, μ is the mean, and σ is the standard deviation.

For x = 46:

z₁ = (46 - 60) / 14

z₁ = -1

For x = 74:

z₂ = (74 - 60) / 14

z₂ = 1

Using the Excel functions:

=NORM.S.DIST(-1) and =NORM.S.DIST(1)

The probabilities are 0.1587 and 0.8413 respectively.

Now, we want the proportion of measurements between z₁ and z₂, which is:

Proportion = 0.8413 - 0.1587

                  ≈ 0.6826

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Final answer:

Using the Empirical Rule for a normal distribution, approximately 68% of the measurements would fall between 46 and 74, as this range lies within one standard deviation above and below the mean of 60 in a distribution with a standard deviation of 14.

Explanation:

To find the proportion of measurements between 46 and 74 in a distribution with a mean of 60 and a standard deviation of 14, we can use the Empirical Rule, assuming the distribution is normal (bell-shaped). This rule states that approximately 68% of the data lies within one standard deviation of the mean, 95% within two, and more than 99% within three.

In this case, 46 is one standard deviation below the mean (60 - 14), and 74 is one standard deviation above the mean (60 + 14). So, we would expect approximately 68% of the measurements to lie between 46 and 74.

This is because the data is likely to be distributed symmetrically around the mean in a normal distribution, and the range given includes measurements falling within one standard deviation from the mean.

Option B:

[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]

Solution:

Given sound level = 120 decibel

To find the intensity of a fire alarm:

[tex]$\beta=10\log\left(\frac{I}{I_0} \right)[/tex]

where [tex]I_0=1\times10^{-12}\ \text {watts}/ \text m^2}[/tex]

Step 1: First divide the decibel level by 10.

120 ÷ 10 = 12

Step 2: Use that value in the exponent of the ratio with base 10.

[tex]10^{12}[/tex]

Step 3: Use that power of twelve to find the intensity in Watts per square meter.

[tex]$10^{12}=\left(\frac{I}{I_0} \right)[/tex]

[tex]$10^{12}=\left(\frac{I}{1\times10^{-12}\ \text {watts}/ \text m^2} \right)[/tex]

Now, do the cross multiplication,

[tex]I=10^{12}\times1\times\ 10^{-12} \ \text {watts}/ \text m^2}[/tex]

[tex]I=1\times\ 10^{12-12} \ \text {watts}/ \text m^2}[/tex]

[tex]I=1\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]

[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]

Option B is the correct answer.

Hence [tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex].

Final answer:

To find the intensity of a sound at 120 dB, we use the formula SIL = 10 log(I / I0). With I0 as 10⁻¹² W/m², we find that I = 1.0 x 10⁰ W/m², corresponding to choice B.

Explanation:

To determine the intensity of a fire alarm that has a sound level of 120 decibels (dB), we use the relationship between sound intensity level and intensity in watts per meter squared (W/m²). The formula to convert decibel level to intensity is:

SIL = 10 log(I / I0)

Where SIL is the sound intensity level in decibels, I is the intensity of the sound, and I0 is the reference intensity, usually taken as 10⁻¹² W/m², the threshold of human hearing. To find the unknown intensity I, we can rearrange the formula:

I = I0 × 10(SIL/10)

For a sound level of 120 dB, the calculation would be:

I = 10⁻¹² W/m² × 10¹²⁰/¹⁰

I = 10⁻¹² W/m² × 10¹²

I = 1.0 × 10⁰ W/m²

Therefore, the correct answer is B. 1.0 x 10⁰ watts/m².

Answer:

i

Step-by-step explanation:

There has to be a mod 26 operation for c, since the alphabet only has 26 letters.

But otherwise

[tex]c = E(p+k) = p + 9 \mod{26}[/tex]

p is the initial position of the letter. So for example, a = 0, b = 1, c = 2, q = 16, z = 25...

9 is by how much the encrypted character will change relative to the original.

We want to cipher the letter q.

So p = 25.

[tex]c = E(p+k) = p + 9 \mod{26}[/tex]

[tex]c = 25 + 9 \mod{26}[/tex]

[tex]c = 34 \mod{26}[/tex]

The remainder of 34 divided by 26 is 8.

So

[tex]c = 8[/tex]

Which means that q is encrypted as 'i'.

In a Caesar Cipher system, when the encryption rule is shifting by 9 places, the letter 'q' is encrypted as 'c'.

In a Caesar Cipher system, the letter 'q' is encrypted using the formula:

c = E(p + k)

Where:

c is the ciphertext (the encrypted letter).

p is the plaintext (the original letter).

k is the encryption key.

You mentioned that the encryption is done with the rule c = p + 9. In this case, k is 9.

To encrypt the letter 'q', you would substitute p with 'q' in the formula:

c = 'q' + 9

Adding 9 to 'q' in the alphabet results in:

'q' + 9 = 'z'

So, the letter 'q' is encrypted as the letter 'z' using this Caesar Cipher system.

Therefore, the encrypted letter for 'q' in a Caesar Cipher system with a shift of 9 is 'c'.

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Answer:

x=e-1 0r 1.71828

Step-by-step explanation:

Step-by-step explanation:

[tex]In(x + 3)^{3} = 3 \\ \therefore \: 3In(x + 3) = 3 \:\\ \therefore \: In(x + 3) = \frac{3}{3} \: \\ \therefore \: In(x + 3) = 1\\ \therefore \: In(x + 3) = In \: 10\\..( \because \: In \: 10 = 1) \\ \therefore \:x + 3 = 10 \\ \therefore \:x = 10 - 3 \\ \: \: \: \: \: \huge \red{ \boxed{\therefore \:x = 7}}[/tex]

Answer: the bill before 6% sales tax is $92.85

Step-by-step explanation:

Let x represent the bill before the 6% sales tax.

It means that you paid 6% tax on x and the amount of tax paid would be

6/100 × x = 0.06 × x = 0.06x

Total amount that you paid for the meal including the 6% tax would be

x + 0.06x = 1.06x

If you spent $98.42 on the meal after the 6% tax, it means that

1.06x = 98.42

Dividing the left hand side and the right hand side of the equation by 1.06, it becomes

1.06x/1.06 = 98.42/1.06

x = $92.85

Final answer:

The hypothesis test to determine the mean annual premium in Pennsylvania compared to the national mean annual premium is H0: μ ≥ 1,503 and Ha: μ < 1,503. The point estimate of the difference between the mean annual premiums is -$78.

Explanation:

(a) To determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium, we need to develop a hypothesis test. The null hypothesis (H0) states that the mean annual premium in Pennsylvania is greater than or equal to the national mean annual premium. The alternative hypothesis (Ha) states that the mean annual premium in Pennsylvania is less than the national mean annual premium. Therefore, the correct answer is:

H0: μ ≥ 1,503
Ha: μ < 1,503

(b) The point estimate in dollars of the difference between the mean annual premium in Pennsylvania and the national mean is calculated by subtracting the national mean annual premium ($1,503) from the mean annual premium in Pennsylvania ($1,425). Therefore, the point estimate is $1,425 - $1,503 = -$78.

Answer:

(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more = 0.0316.

(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 0.9684 .

Step-by-step explanation:

We are given that 20% of all packages from Amazon are delivered by UPS, from which 12% of the packages that are delivered by UPS weighs 2 lbs or more and 8% of the packages that are not delivered by UPS weighs less than 2 lbs.

Firstly Let A = Package from Amazon is delivered by UPS.

                B = Packages that are delivered by UPS weighs 2 lbs or more.

So, P(A) = 0.2  and P(A') = {Probability that package is not delivered by UPS}

                                P(A') = 1 - 0.2 = 0.8

P(B/A) = 0.12 {means Probability that package weight 2 lbs or more given it

                      is delivered by UPS}

P(B'/A') = 0.08 [means Probability that package weight less than 2 lbs given

                         it is not delivered by UPS}

Since, P(B/A) = [tex]\frac{P(A\bigcap B)}{P(A)}[/tex]   ,    [tex]P(A\bigcap B)[/tex] = P(B/A) * P(A) = 0.12 * 0.2 = 0.024 .

Also P(B) { Probability that package weight 2 lbs or more} is given by;

So, P(B) = [tex]P(B\bigcap A) + P(B\bigcap A')[/tex] = P(B/A) * P(A) + P(B/A') * P(A')

             = 0.12 * 0.2 + 0.92 * 0.8 { Here P(B/A') = 1 - P(B'/A') = 1 - 0.08 = 0.92}

             = 0.76

(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more  is given by P(A/B);

  P(A/B) =   [tex]\frac{P(A\bigcap B)}{P(B)}[/tex] =  [tex]\frac{0.024}{0.76}[/tex] = 0.0316

(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 1 - P(A/B) = 1 - 0.0316 = 0.9684 .            

Answer:

A) C(d,m) = 40 + 55d + 0.13m

B) $448

Step-by-step explanation:

Let 'd' be the number of days and 'm' the number of miles driven.

A) The cost function that describes a fixed amount of $40, added to a variable amount of $55 per day (55d) and a variable amount of 13 cents per mile (0.13m) is:

[tex]C(d,m) = 40 +55d +0.13m[/tex]

B) If  d = 5 and m =600, the total cost is:

[tex]C(5,600) = 40 +55*6 +0.13*600\\C(5,600)=\$448[/tex]

The cost is $448.

The formula for the cost, C of renting a car as a function of the number of days, d, and the number of miles driven, m is C f(d, m) = 40 + 55d + 0.13m

Given:

Application fee = $40

cost per day = $55

cost per mile = $0.13

let

Total cost = C

Number of days = d

Number of miles = m

Total cost, C = 40 + 55d + 0.13m

C f(d, m) = 40 + 55d + 0.13m

= 40 + 55(5) + 0.13(600)

= 40 + 275 + 78

= $393

Therefore, the cost of renting the car given the number of days is $393

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Answer:

the probability of choosing a king for the second card drawn is 3/51 , if the first card, drawn without replacement, was a king

Step-by-step explanation:

defining the variable F= choosing a king in the first drawn , then the probability is

P(F)= 4/52

then using the theorem of Bayes for conditional probability and denoting the event S= choosing a king in the second drawn , then

P(S/F)= P(S∩F)/P(F)

where

P(S∩F) = probability of choosing a king in the first drawn and second drawn = 4/52* 3/51

P(S∩F) =probability of choosing a king in the second drawn given that a king was chosen in the first drawn

then

P(S/F)= P(S∩F)/P(F) =  4/52* 3/51 / 4/52 = 3/51

The probability of choosing a king for the second card drawn is 1/17 or approximately 0.0588.

To find the probability of choosing a king for the second card drawn, given that the first card was a king, we need to consider the total number of cards and the number of kings remaining after the first card was drawn.

After the first king is drawn, there are 51 cards left in the deck, with 3 remaining kings. Therefore, the probability of choosing a king for the second card drawn is 3/51, which can be simplified to 1/17 or approximately 0.0588.

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Step-by-step explanation:

There are 16 balls total.  Since the balls are replaced after selection, the probability that the ball is green is 9/16 each time.  The probability that a green ball is selected 6 times is:

P = (9/16)^6

P = 0.032

Answer:

495

Step-by-step explanation:

Using combination without repetition, the formular is given by n!/(r!(n-1)!

= (12 x11 ×10 ×9) /4 !

=11880/24

=495

From the group of 16 students with 5 math majors, 3 physics major and 8 engineering majors a team of 4 students where there has been a student from each majors total no. of teams that can be formed is 111.

Combination is selecting some things in which order does not matter.

According to the given question

A class of 16 students contains 5 Math majors, 8 Engineering majors and 3 Physics majors. A group of four students from the class is to be selected to form a team for an academic competition.

Here three cases will be formed as group is formed by 4 students and the no. of majors is 3.

First case when 2 math majors, 1 physics major and 1 engineering major is chosen from 16 students.

= ⁵C₂ + ³C₁ + ⁸C₁

= 8 + 5 + 3

= 16.

Second case when 1 math major, 2 physics major and 1 engineering major is chosen from 16 students.

= ⁵C₁ + ³C₂ + ⁸C₁

= 8 + 20 + 3

= 31.

In the third case when 1 math major, 1 physics major and 2 engineering major is chosen from 16 students.

= ⁵C₁ + ³C₁ + ⁸C₂

= 56 + 5 + 3

= 64.

∴total no of teams that can be formed is

= 16 + 31 + 64

= 111.

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Answer:

x

=

1

+

i

√

79

8

,

1

−

i

√

79

8

Step-by-step explanation:

Answer:

e.

Step-by-step explanation:

The parameter is used to measure the population or in other words the measurement taken from a population is known as parameter. Here, the population will be all the co-workers that use company's healthcare. The percentage calculated for all co-workers that use company's healthcare will be  the measurement calculated from population. Thus, the parameter will be the percentage of all the co-workers that use the company's healthcare.

Final answer:

The parameter Ralph is trying to estimate is 'e. the percentage of all coworkers that use the company's healthcare,' which reflects the actual percentage for the entire population of coworkers, not just the surveyed sample.

Explanation:

The question posed by the student is concerned with identifying the parameter involved in a statistical study. In the scenario provided, Ralph wants to estimate the percentage of coworkers that use the company's healthcare by surveying a random sample of 200 coworkers. The parameter, in this case, is the actual percentage of all coworkers at the company who use the company's healthcare, because it describes the entire population that is being studied, not just the sample that was surveyed. Therefore, the correct answer is e. the percentage of all coworkers that use the company's healthcare.

A parameter is a summary measure that describes an entire population, whereas a statistic is a summary measure that describes a sample taken from the population. An example that illustrates this distinction is an insurance company determining the proportion of all medical doctors who have been involved in malpractice lawsuits by surveying a random sample of 500 doctors. The true proportion of all doctors who have faced lawsuits is a parameter, while the proportion calculated from the sample of 500 doctors is a statistic.

We can take the square root of both sides, adding a plus/minus sign of the right hand side:

[tex]\sqrt{(y+150)^2}=\pm\sqrt{50}\iff y+150 = \pm\sqrt{50}[/tex]

Then, we subtract 150 from both sides:

[tex]y=\pm\sqrt{50}-150[/tex]

So, the two solutions are

[tex]y_1 = \sqrt{50}-150,\quad y_2 = -\sqrt{50}-150[/tex]

Answer:

Are the events A and B disjoint? Yes

Step-by-step explanation:

Disjoint events are those events that cannot occur at the same time, i.e. for events X and Y to be disjoint, [tex]P(X\cap Y)=0[/tex].

The event A is defined as the number of students whose age is at most 32.

And event B is defined as the number of students whose age is at least 37.

The events A and B are disjoint events.

The sample space for the intersection of these two events is:

Sample space of (A ∩ B) = 0

As there are no common terms in both the sample.

Hence proved, events A and B are disjoint.

Answer: Vector b is not a linear combination

Step-by-step explanation:

First of all we put the vectors in terms of different variables, such as:

a1(1,-2,0)=(a,-2a,0);

a2(0,1,3)=(0,b,3b);

a3(6,-6,18)=(6c,-6c,18c);

To know that a vector is a linear combination we need to express it like a sum of other different vectors.

(2,-2,6)=(a,-2a,0)+(0,b,3b)+(6c,-6c,18c)

(2,-2,6)=(a+0+6c,-2a+b-6c,0+3b+18c)

We express this sum like a system of equations.

a+6c=2

-2a+b-6c=-2

3b+18c=6

We solve this system of equations and we can note that the system don't have a solution, so the vector b is not a linear combination of a1, a2, and a3.

Upon forming a system of linear equations and solving, a solution would imply that vector b is indeed a linear combination of vectors a1, a2, and a3. The observed placement of pivots in the corresponding echelon matrix backs this conclusion.

In this question, you are asked to determine if vector b is a linear combination of vectors a1, a2, and a3. A vector is a linear combination of others if it can be written as a weighed sum of those vectors. To solve this problem, we need to form a system of linear equations based on the vectors and solve this system. If all of the coefficients can be expressed as real numbers, it means that the vector b is a linear combination of a1, a2, and a3.

In this case, our system of equations looks like this:

x∗a1 + y∗a2 + z∗a3 = b

In matrix form it can be written as:

|1 0 6|x| = |2|, |-2 1 -6|y| = |-2|, |0 3 18|z| = |6|.

Solve this system through methods like Gauss-Jordan elimination or row reduction. The pivots in the corresponding echelon matrix should be in the first entry in the first column, the second entry in the second column, and the third entry in the third column.

This suggests that vector b can indeed be a linear combination of vectors a1, a2, and a3.

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So, the probabilities are: (a) 37/64, (b) 3/64, (c) 27/64, (d) 57/64

In each spin, there are four possible outcomes (nun, gimel, hei, shin), and each outcome is equally likely.

(a) Probability of getting at least one nun:

The probability of getting no nuns in a single spin is 3/4. So, the probability of getting no nuns in three spins is [tex](3/4)^3[/tex]. Therefore, the probability of getting at least one nun is 1 - [tex](3/4)^3[/tex].

Probability of getting at least one nun:

1 - [tex](3/4)^3[/tex] = [tex]1-\frac{27}{64}=\frac{37}{64}[/tex] = 0.58

(b) Probability of getting exactly 2 nuns:

The probability of getting a nun in a single spin is 1/4. So, the probability of getting exactly 1 hei in three spins is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex].

= [tex]3 \times \frac{1}{16} \times \frac{3}{4}=\frac{3}{64}[/tex] = 0.05

(c) Probability of getting exactly 1 hei:

The probability of getting a hei in a single spin is 1/4. So, the probability of getting exactly 1 hei in three spins is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex]

= [tex]3 \times \frac{1}{4} \times \frac{9}{16}=\frac{27}{64}[/tex] = 0.42

(d) Probability of getting at most 2 gimels:

The probability of getting 0 gimels is [tex](3/4)^3[/tex]. The probability of getting 1 gimel is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex]. The probability of getting 2 gimels is [tex]\left(\begin{array}{l}\frac{3}{2} \\\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)[/tex].

Add these probabilities to get the total probability.

[tex]\left(\frac{3}{4}\right)^3+\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2+\left(\begin{array}{l}\frac{3}{2} \\\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)[/tex]

[tex]=\frac{27}{64}+\frac{27}{64}+\frac{3}{64}=\frac{57}{64}[/tex] = 0.9

​

Calculating probabilities of specific outcomes when spinning a dreidel multiple times.

Dreidel Probability Calculations:

Volume of the cone is 117.5 π ft³

Step-by-step explanation:

Lateral area of the cone = πrs

s is the slant height = 15 ft

From the above formula, we can find the radius as, 5 ft.

Volume of the cone = π r² h/3

s = √ (5²+ h²)

Squaring on both sides, we will get,

s² = 15² = (5² + h²)

 15² - 5² = h²

225 - 25 = 200 = h²

h = √200 = 14.1 ft

Volume = π × 5² × 14.1  / 3 = 117.5 π ft³

Answer:

Kelly will owe $1320.92 to the fabulous fence company.

Step-by-step explanation:

There is a cost related to the number of hours and a cost per feet. So the total cost is:

[tex]T = C_{h} + C_{f}[/tex]

In which [tex]C_{h}[/tex] is the cost related to the number of hours and [tex]C_{f}[/tex] is the cost related to the number of feet.

Cost per hour

Each hour costs $15.57.

They work for 6 hours total. So

[tex]C_{h} = 15.57*6 = 93.42[/tex]

Cost per feet

Each feet costs $3.25.

Kelly needs 350 feet. So

[tex]C_{f} = 350*3.25 = 1137.5[/tex]

The total cost is:

[tex]T = C_{h} + C_{f} = 93.42 + 1137.5 = 1230.92[/tex]

Kelly will owe $1320.92 to the fabulous fence company.

Answer:

The sample space for the experiment is {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.

Step-by-step explanation:

Consider the provided information.

Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.

The sample space for the experiment is:

{(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}

Hence, the possible outcomes are {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.

Answer:

A:   d. Yes, the program is statistically significant because the results are likely to occur by chance.

B: d. Yes, the program is practically significant because the amount of lost weight is large enough to be considered practically significant.

Step-by-step explanation:

A likely event is the one which is likely to occur or show the same results.

An unlikely event is one in which there's a chance of not getting the desired result or may not show the same results.

In the given scenario the program is statistically significant because even if the 28 % results occur likely by chance the rest  72 % will show some desired results.

In part B the program is statistically practical as the remaining 72 % represent a significant stats.

Let [tex]t[/tex] be the number of minutes.

The first tank starts with 1500 liters, and loses 4 liters per minute, so after [tex]t[/tex] minutes there will be

[tex]1500-4t[/tex]

liters of water.

The second tank is filled at 6 liters per second, i.e.

[tex]6\times 60=360[/tex] liters per minute.

So, there will be

[tex]300+360t[/tex]

liters of water in the second tank after [tex]t[/tex] minutes.

The two quantities will be equal when

[tex]1500-4t=300+360t \iff 1200=364t \iff t=\dfrac{1200}{364}\approx 3.3[/tex]

so, approximately, after 3.3 minutes.

Answer: it will take about 3.3 minutes for both tanks to have the same amount of water.

Step-by-step explanation:

Let x represent the number of minutes it will take both tanks to have same amount of water.

A water tank has 1,500 liters of water. It has a leak, losing 4 liters per minute. This means that in x minutes, the volume of water in the tank would be

1500 - 4x

At the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per second. Converting 6 liters per second to minutes, it becomes 60 × 6 = 360 liters per minute. This means that in x minutes, the volume of water in the tank would be

300 + 360x

For both tanks to have same amount of water, then

1500 - 4x = 300 + 360x

360x + 4x = 1500 - 300

364x = 1200

x = 1200/364 = 3.3 minutes

Answer: t = - 5 ∈ [tex]I_{1}[/tex] = ( -∞ , -4 )

Step-by-step explanation:

The standard form of O.D.E is written as :

[tex]y^{1}[/tex] + [tex]p(t) = g(t)[/tex]

Equation given :

[tex](16-t^{2} )y^{1}[/tex] + [tex]9ty[/tex] = [tex]5t^{2}[/tex] ,       [tex]y(-5) = 1[/tex]

The first thing to do is to write the O.D.E in standard form , that is we will divide through by [tex]16 - t^{2}[/tex] , so we have

[tex]y^{1} + \frac{9ty}{16-t^{2}}=\frac{5t^{2}}{16-t^{2}}[/tex]

With this , we can see that [tex]p(t)[/tex] and [tex]g(t)[/tex] are both continuous in the same domain. Therefore , the intervals are :

[tex]I_{1}[/tex] = ( -∞ , -4 )

[tex]I_{2}[/tex] = ( - 4 , 4 )

[tex]I_{3}[/tex] = ( 4 , -∞ )

recall that y(−5) = 1 , then t = -5

This means that :

t = - 5 ∈ [tex]I_{1}[/tex] = ( -∞ , -4 )

Answer:

The outlier for this case would be (19.37 , 337) since this point is far away from the others and this point probably strongly influences the correlation value.

What is the correlation with this point? (Round your answer to two decimal places.)

> cor(x,y)

[1] 0.34

What is the correlation without this point? (Round your answer to two decimal places.)

> cor(x1,y1)

[1] 0.79

Calculate the regression line with the outlier.

y = 35.2 x-523.9

Calculate the regression line without the outlier.

y = 77.9 x -1422.3

Step-by-step explanation:

We have the following data:

Isotope %: 19.90, 20.71, 21.63, 19.84, 20.80, 21.63, 19.46, 20.86, 21.19,20.20, 21.28, 19.37 (representing X)

Silicon : 85,152,226,106,263,233,114,265,186,139,298,337 (representing Y)

Find the single outlier in the data. This point strongly influences the correlation. What is the correlation with this point? (Round your answer to two decimal places.)

We can use the scatter plot in order to see any potential outlier. With the following R code:

> x<-c(19.90, 20.71, 21.63, 19.84, 20.80, 21.63, 19.46, 20.86, 21.19,20.20, 21.28, 19.37)

> y<-c(85,152,226,106,263,233,114,265,186,139,298,337)

> plot(x,y, main="Scatter plot Silicon vs Isotope")

And we can see the plot on the figure attached.

The outlier for this case would be (19.37 , 337) since this point is far away from the others and this point probably strongly influences the correlation value.

What is the correlation with this point? (Round your answer to two decimal places.)

> cor(x,y)

[1] 0.34

What is the correlation without this point? (Round your answer to two decimal places.)

> x1<-x[-12]

> x1

[1] 19.90 20.71 21.63 19.84 20.80 21.63 19.46 20.86 21.19 20.20 21.28

> y1<-y[-12]

> y1

[1]  85 152 226 106 263 233 114 265 186 139 298

> cor(x1,y1)

[1] 0.79

As we can see the correlation changes significantly without the outlier.

c) Is the outlier also strongly influential for the regression line? Calculate the regression line with the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ? x ( )

We can calculate the regression line with the following R code

> linearmod1<-lm(y~ x)

> linearmod1

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept)            x  

    -523.9         35.2  

So our equation would be: y = 35.2 x-523.9

Calculate the regression line without the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ?( ) x

> linearmod2<-lm(y1~x1)

> linearmod2

Call:

lm(formula = y1 ~ x1)

Coefficients:

(Intercept)           x1  

  -1422.26        77.85  

The new equation would be y = 77.9 x -1422.3

So as we can see the outlier also changes significantly the estimation for the slope and the intercept of the linear model

To answer this question one would need to identify the outlier in the data then calculate both the correlation and regression lines with and without this outlier. Changes in these calculations can demonstrate the influence of the outlier.

This question involves finding statistical outliers and calculating correlation and regression lines in a dataset. The outlier in a dataset is a data point that is remarkably distinct from the rest of the data. As your question doesn't provide a clear dataset, it's impossible to identify the outlier clearly.  However, once identified, to determine if the point strongly influences the correlation you would find the correlation with the outlier and without the outlier and compare these two values.

If the

correlation

drastically changes when the outlier is removed, it can confirm that the point is strong influencing the correlation. The

regression line

could be calculate using the standard formulas for slope and intercept, both with and without the outlier in the data. If the regression line meaningfully shifts after removing the outlier, it indicates that outlier is impactful to the regression line as well. This analytical work typically requires skills in interpreting scatterplots and using statistical programs or calculators.

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